PinT2018: /2018/5/4 ROSCOFF Convergence Acceleration of the PinT Integration of Advection Equation using Accurate Phase Calculation Method Mikio Iizuka Kyushu University, RIIT （ Research Institute for Information Technology ） Kenji Ono Kyushu University, RIIT （ Research Institute for Information Technology ） RIKEN Advanced Institute for Computational Science, The University of Tokyo, Institute of Industrial Science, Japan 1
Acknowledgement This work was supported in part by MEXT ( Ministry of Education, Culture, Sports, Science and Technology ) as a priority issue (“Development of Innovative Design and Production Processes that Lead the Way for the Manufacturing Industry in the Near Future” ) to be tackled by using Post ‘K’ Computer. This research used computational resources of the K computer provided by the RIKEN Advanced Institute for Computational Science through the HPCI System Research project (Project ID:hp170238). 2
Outline ⚫ Int ntro rodu duct ction on ⚫ How ow do do we we de deve velop op ve very a acc ccurate ph phase ca calcl clation on metho hod d ? ⚫ Method imp mproving t the c cal alculat ation me method of of advection ad ⚫ Check impacts of conventional methods improvement ⚫ Parareal ca calcu culation on ⚫ Summary and nd Future W Wor ork 3
Intro In rodu duction 4
Basis direction of our PinT research Engineering Viewpoint (CFD etc) our PinT research 5
Basis direction of our PinT research Loves widely usable method, especially Engineering the method usable to hyperbolic PDEs Viewpoint (CFD etc) our PinT research 6
Basis direction of our PinT research Loves widely usable method, especially Engineering the method usable to hyperbolic PDEs Viewpoint (CFD etc) Now, mainly, Advection eq. our PinT research 7
Basis direction of our PinT research Loves widely usable method, especially Engineering the method usable to hyperbolic PDEs Viewpoint Love simple. → Pararael method is better. (CFD etc) do not like complex platform or frame work. Now, mainly Advection eq. our PinT research 8
Basis direction of our PinT research Loves widely usable method, especially Engineering the method usable to hyperbolic PDEs Viewpoint Love simple. → Pararael method is better. (CFD etc) do not like complex platform or frame work. Would you change your solver to incorporate your cord in PinT-Frame? Now, mainly Advection eq. our PinT research 9
Basis direction of our PinT research Loves widely usable method, especially Engineering the method usable to hyperbolic PDEs Viewpoint Love simple. → Pararael method is better. (CFD etc) do not like complex platform or frame work. Would you change your solver to incorporate your cord in PinT-Frame? NO! NO! NO! NO! Now, mainly NO! Advection eq. NO! our PinT research 10
Basis direction of our PinT research Loves widely usable method, especially Engineering the method usable to hyperbolic PDEs Viewpoint Loves simple. → Pararael method is better. (CFD etc) do not like complex platform or frame work. Now, mainly Advection eq. Parareal our PinT research 11
Basis direction of our PinT research Loves widely usable method, especially Engineering the method usable to hyperbolic PDEs Viewpoint Loves simple. → Pararael method is better. (CFD etc) do not like complex platform or frame work. Have no interest in parallel method or HPC! → Now, mainly Have an interest in Advection eq. numerical method at least. Parareal our PinT research 12
Basis direction of our PinT research Loves widely usable method, especially Engineering the method usable to hyperbolic PDEs Viewpoint Love simple. → Pararael method is better. (CFD etc) do not like complex platform or frame work. Have no interest in parallel method or HPC! → Now, mainly Have an interest in Advection eq. numerical method at least. Parareal our PinT Numerical research method of advection 13 2
Issue of the parareal method for hyperbolic PDEs: Parareal convergence ODE Relatively good Parabolic PDE DE BAD Hyperbolic PDE 14
Why so bad? *M. Gander and M. Petcu, Analysis of a Krylov subspace enhanced parareal algorithm for linear problems, ESAIM Proc, 25, (2008), 114-129. ⚫ Hyp yperbolic PDEs r represents ts wav ave p phenomena. a. ⚫ If th there i is Phas ase D Difference between fine and coarse solver’s t → Os result Oscillat ations ap appear ars at at th the e edge of ti time s slice. ⚫ That at g give ves dam amag age th the conve vergence of th the par arar areal al meth thod. Our challenge 3 Reducing the phase difference between fine/coarse solvers 15
Example of oscillations in parareral iteration for advection equation Profile of Φ at t=0.5 after 10 iteration with time-coarsening ratio Rfc=25 without relaxation of iteration Fine/coarse solver :TVD/CN Fine/coarse solver CIP3rd method Φ 400 5 Φ 200 C=1 0 0 -200 -5 x 0 0.5 1 1.5 2 -400 x 0 0.5 1 1.5 2 x step step Oscillation amplitudes much depend on numerical integration methods (may be accuracy of pahse calcultion). 16
We e stu tudi died ed t the he impa pact ct of of ph phas ase e di differ eren ence ce on on th the e pa parareal co conv nvergenc nce using ng m mos ost simpl ple p prob oblem. (a) Most simple problem: This gives the exact phase to → Simple harmonic motion fine/coarse solver. → Simplest hyperbolic PDE We tried to check the effect of phase difference by adding the erro to coarse solver by value ε. (b) Time integrator: Modified Newmark- β Method This method can give the exact phase for the simple harmonic motion independent on time step width by the modified δt’, δT’. Fine solver Coarse solver This results shows that very very small phase difference causes the convergence difficulty. 17
We e stu tudi died ed t the he impa pact ct of of ph phas ase e di differ eren ence ce on on th the e pa parareal co conv nvergenc nce using ng m mos ost simpl ple p prob oblem. (a) Most simple problem: This gives the exact phase to → Simple harmonic motion fine/coarse solver. → Simplest hyperbolic PDE We tried to check the effect of phase difference by adding the erro to coarse solver by value ε. (b) Time integrator: Modified Newmark- β Method Therefore, we focus on development of This method can give the exact phase the me th meth thod th that at reduc uce ph phas ase for the simple harmonic motion independent on time step width by the modified δt’, δT’. difference ce betw tween fine/co coar arse se Fine solver Coarse solver so solv lver by ap appl ply th the very ac accu curat ate ph phas ase ca calc lcla lati tion me meth thod to to This results shows that very very small phase fine/co coar arse se so solv lver. difference causes the convergence difficulty. 6 18
How ow do do we we de deve velop op ve very ry ac accu curat rate e ph phas ase e ca calcl clat atio ion n me meth thod od ? 19
This research approach:1 Approach based Approach based on the engineering method on mathematics of parareal method Dramatically Improving the conventional calculation method of advection equation to increase the phase accuracy Speedup Conventional method Residual Ｉ mprovement method ⚫ δT δt ⚫ Reduce the time span: T → Σ^{nc}_{l=1} T_{l} ⚫ etc 20
Methods overview of advection term calculation Methods that are tried in this study Conventional 1st: CIP scheme improvement : advecting the main method much phase information by the gradient and Stabilization: curveutre of value Φ. numerical damping 2 nd : STRS scheme: achieving the stabilization Accuracy: and error elimination using “space and time Space and time reversal symmetry “ base on the physics. higher order terms Advecting only 3rd: Hybrid of CIP method and STRS scheme amplitude of variables There is a limit in the us Main issue : Gap of phase accuracy between fine and coarse solver not enough Hybrid: STRS-CIP not yet success There is a limit in the use. 21 8
This research approach:2 Conventional methods of advection equation lose phase accuracy for high grid based wave number waves except CIP3rd method. Dispersion relations numerical calculation for advection equation. Grid based wave number : k=2π/λ= 2π/m/Δx 22
Simple and Typical Benchmark Problem including high grid based wave number waves Sin wave advection problem Step wise advection problem with very rough grids Φ(x) Analytical 1 CIP3rd CIP5th 1 0.8 0.8 0.6 0.6 0.4 Step Curved 0.2 Step Curved 0 0.4 -0.2 -0.4 0.2 -0.6 -0.8 -1 0 1.8 1.85 1.9 1.95 2 0 0.5 1 1.5 2 2.5 3 x Most simple problem: Most tough problem: Including high grid Including broad and high based wave number grid based wave number one wave. waves 23
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